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Scaling law for the size dependence of a finite-range quantum gas

Luigi Delle Site, Carsten Hartmann

Abstract

In a recent work [Reible et al., Phys. Rev. Res. 5, 023156, 2023], it has been shown that the mean particle-particle interaction across an ideal surface that divides a system into two parts, can be employed to estimate the size dependence for the thermodynamic accuracy of the system. In this work we propose its application to systems with finite range interactions that models a dense quantum gases and derive an approximate size-dependence scaling law. In addition, we show that the application of the criterion is equivalent to the determination of a free energy response to a perturbation. The latter result confirms the complementarity of the criterion to other estimates of finite-size effects based on direct simulations and empirical structure or energy convergence criteria.

Scaling law for the size dependence of a finite-range quantum gas

Abstract

In a recent work [Reible et al., Phys. Rev. Res. 5, 023156, 2023], it has been shown that the mean particle-particle interaction across an ideal surface that divides a system into two parts, can be employed to estimate the size dependence for the thermodynamic accuracy of the system. In this work we propose its application to systems with finite range interactions that models a dense quantum gases and derive an approximate size-dependence scaling law. In addition, we show that the application of the criterion is equivalent to the determination of a free energy response to a perturbation. The latter result confirms the complementarity of the criterion to other estimates of finite-size effects based on direct simulations and empirical structure or energy convergence criteria.
Paper Structure (12 sections, 4 theorems, 69 equations, 1 figure)

This paper contains 12 sections, 4 theorems, 69 equations, 1 figure.

Table of Contents

  1. Introduction
  2. A mathematically sound criterion to estimate thermodynamic accuracy as a function of the system size
  3. Relative free energy difference as a quality measure
  4. Quantum gas with finite range potential
  5. Simple scaling law
  6. Interdomain energy as a thermodynamic response function
  7. Particle insertion: $E_{\rm intd}$ and the chemical potential
  8. Alchemical transformation: free energy perturbation method
  9. An illustrative calculation using Gaussians
  10. Conclusions
  11. Concavity and differentiability of the interface free energy
  12. Matrix calculations

Key Result

Theorem 1

It holds where $\mathbf{E}_f[\cdot]$ or $\mathbf{E}_{f_{1}f_{2}}[\cdot]$ are the expectations with respect to the probability density functions $f$ or $f_1f_2$.

Figures (1)

  • Figure 1: Interface free energy and its upper and lower bounds for two weakly coupled linear oscillator chains, each of which consists of $n=m=100$ particles. Here $\varepsilon$ is the coupling parameter that models the strength of the bilinear interactions between the 100th particle and the 101st particle.

Theorems & Definitions (9)

  • Definition 1: Interface free energy
  • Theorem 1: Two-sided Bogoliubov inequality
  • Theorem 2
  • proof
  • Definition 2: Cumulant generating function
  • Lemma 3
  • proof
  • Lemma 4
  • proof